University of Central Florida



Understanding Projectile Motion

Quadratic Applications

Description

Here, you are exposed to two situations of motion and asked to explain the differences, if any, in the equations of motion.

Objectives (Lessons to be learned)

• You should be able to understand that if something is shot up into the air (assuming no thrusting components such as an engine) the only thing it can do is start to come down due to gravity.

• You should understand that this basically means that the curve of the path is quadratic, NOT linear.

• You should understand the concept of gravity, velocity, and distance traveled.

Sunshine State Standards/Benchmarks

• MA.D.1.4.1 I can describe, analyze, and generalize relationships, patterns, and functions using words, symbols, variables, tables, and graphs.

• MA.D.1.4.2 I can determine the impact when changing parameters of given functions such as finding the change in area of a circle when the radius is doubled.

Bodies of Knowledge (Approved September 2007)

• MA.912.A.2.13 I can solve real-world problems involving relations and functions.

• MA.912.A.4.10 I can use polynomial equations to solve real-world problems.

• MA.912.A.7.1 I can graph quadratic equations with and without graphing technology.

• MA.912.A.7.2 I can solve quadratic equations over the real numbers by factoring, and by using the quadratic formula.

• MA.912.A.7.8 I can use quadratic equations to solve real-world problems

• MA.912.A.7.9 I can solve optimization problems.

• MA.912.A.7.10 I can use graphing technology to find approximate solutions of quadratic equations.

• MA.912.C.3.4 I can find local and absolute minimum and maximum points

Relevance

This activity provides a means of understanding what the physical meaning of a quadratic equation really is. It gives a good example of how gravity physically acts on an object trying to overcome it.

Tools Needed

• Pencil

• Paper

• TI-84 Plus Silver Edition graphing calculator

• Calculator-Based Ranger (CBR 2)

• Basketball

Assignment

Part A

Sam shoots a missile straight up with an initial velocity of 96 ft/s from the ground. Find the height of the object after the following times using d=r*t:

|Time |Height |

|(sec) |(ft) |

|0 |0 |

|1 | |

|2 | |

|3 | |

1. After using the formula d=rt to complete the chart, do the answers seem reasonable? Explain why or why not.

2. What major real-life factor is missing from the formula?

Experimental Activity

Each group should receive a basketball and should follow the instructions below.

• Throw the ball in the air as vertically as possible and observe the motion of the ball.

• Draw what you think the plot of height vs. time is for the motion you just observed.

Why did you draw the graph the way you did? Explain this physically in a sentence or two below.

• Throw the ball again just as you did earlier except this time, using the motion detector, obtain the plot of the motion of the ball using the graphing calculator. Sketch what you see below.

Does this match the plot that you guessed without the calculator and motion detector? Why or why not? Why physically does the plot that the calculator gave you make more sense?

• Now, roll the ball on the floor and draw what you think the plot of horizontal distance vs. time is for the motion you just observed.

Why did you draw the graph the way you did? Explain this physically in a sentence or two below.

• Roll the ball again just as you did earlier except this time, using the motion detector, obtain the plot of the motion of the ball using the graphing calculator. Sketch what you see below.

Does this match the plot that you guessed without the calculator and motion detector? Why or why not? Why physically does the plot that the calculator gives you make sense?

What is the difference in the plots you received from the calculator when the ball was thrown in the air and when it was rolled? Explain the differences mathematically, and physically.

What is the main physical, every day factor that makes the motion of the vertically thrown ball different from the rolled ball?

From the different quadratic equations that each group has, what patterns do you notice about the “a” term, the “b” term and the “c” term? Hint: Try and find the term that would give you 0.5*gravity where gravity is 32.2 feet/s or 9.81 m/s. Is this term the a, b, or c term?

3. In the next table, make a table of values that are more realistic. They don’t have to be exactly correct. I just want to see what you think would make more sense in real life. Explain how you chose the values.

|Time |Height |

|(sec) |(ft) |

|0 |0 |

|1 | |

|2 | |

|3 | |

Part B

Mathematicians and Physicists developed a formula which does take into account gravity.

Projectile Motion Formula*:

h(t)= ho+vot+½ at2

*also called Vertical Motion Formula

Defining our Variables:

h(t)=height after a certain amount of time: example h(2) means height after 2 seconds.

vo=starting vertical velocity or vertical velocity when time =0 seconds

t=time in seconds

a=acceleration (due to gravity on Earth= -32ft/s2 or -9.8 m/s2)

ho=starting height or height when time=0 seconds

1. Why do we have to say starting velocity in this formula instead of just velocity?

2. Why do we have to say starting height instead of just height?

3. Why do we have to say gravity on Earth has a negative value?

4. What part of the Projectile Motion Formula is equivalent to d=rt?

5. What part of the Projectile Motion Formula accounts for gravity?

6. Why does the Projectile Motion Formula have ho as a variable?

Let’s redo the original problem using the Projectile Motion Formula instead of d=rt.

Sam shoots a missile straight up with an initial velocity of 96 ft/s from the ground.

Using our definitions, I have filled in the known (or implied) values. Explain how I knew each answer for this example.

h(t)=h(t) because ___________________________

vo=96 because______________________________

t=t because ________________________________

a= -32 because _____________________________

ho=0 because ______________________________

Rewrite the formula with known values to get h(t)=0+96t+½ (-32)t2

Simplify to obtain: h(t)=96t-16 t2

Now complete the chart and explain why these answers seem more reasonable.

|Time |Height |

|(sec) |(ft) |

|0 |0 |

|1 | |

|2 | |

|3 | |

Part C

Extend the chart for t=4, 5 etc. and stop when it seems reasonable. How did you decide when to stop? __________________________

|Time |Height |

|(sec) |(ft) |

|4 | |

|5 | |

| 6 | |

|7 | |

| | |

| | |

Now graph the data on a coordinate plane. Label the x-axis as time (sec) and label the

y-axis as height (ft). Use graph paper or the blank space next to the table.

Part D

1. How long did it take for the missile to hit the ground?

2. At what time values was the missile on the ground? (2 answers)

3. At what time was the missile at its maximum height?

4. What do you notice when you compare the time values when the missile was at ground level and the time value when the missile was at its maximum height?

This means that if you can find the roots (x-intercepts or t-intercepts) of a quadratic, you can take the average of the roots to find the time when the object will be at its maximum.

Part E

A dancer starts a jump from a 3 foot box with an initial vertical velocity of 8 ft/second. What is the maximum height? When will he reach the stage? Hint: use quadratic formula. Draw a graph show height (ft) vs. time (seconds).

Part F

A word problem about a dancer instructs you to use the function h(t)=5+10t-16t^2. Describe what the 5, 10, and -16 represent in the function.

Part G

A word problem about a dancer instructs you to use the h(t)=5+10t-8t^2. Explain why this is not reasonable.

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